Question

The iterative formula for finding the approximate root of $f(x) = 0$ using Newton-Raphson method is:

• A. \(x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}\)
• B. \(x_n = x_{n+1} + h f(x_n, y_n)\)
• C. \(x_n = \frac{x_{n-1} + x_{n-2}}{f(x_{n-1})}\)
• D. \(x_n = \frac{x_{n-1}f(x_{n-2}) - x_{n-2}f(x_{n-1})}{f(x_{n-1}) - f(x_{n-2})}\)

Hint:

The Newton-Raphson method features a quadratic rate of convergence ($\text{order of convergence } p = 2$), meaning that the number of correct decimal places roughly doubles with each successive iteration step, provided the initial guess is close enough to a simple root.

Answer 1

The Correct Option is A

Concept: The Newton-Raphson Method is a powerful numerical technique used to find successively better approximations to the real roots of a real-valued function $f(x) = 0$. Geometrically, the method approximates the function curve locally by its tangent line. Given an initial guess $x_0$, the tangent line to the curve $y = f(x)$ at the point $(x_0, f(x_0))$ is constructed. The next approximation, $x_1$, is defined as the $x$-intercept of this tangent line.

Step 1: Derivation of the tangent line equation.
Let our current approximation step be denoted by $x_{n-1}$. The point on the curve is $(x_{n-1}, f(x_{n-1}))$, and the slope of the curve at this location is given by the derivative value $f'(x_{n-1})$. The point-slope equation of the tangent line is: \[ y - f(x_{n-1}) = f'(x_{n-1}) \cdot (x - x_{n-1}) \]

Step 2: Finding the $x$-intercept.
The next refined iteration coordinate, $x_n$, occurs where this line intersects the $x$-axis (i.e., setting $y = 0$): \[ 0 - f(x_{n-1}) = f'(x_{n-1}) \cdot (x_n - x_{n-1}) \]

Step 3: Isolate $x_n$.
Dividing both sides by the derivative value $f'(x_{n-1})$, assuming $f'(x_{n-1}) \neq 0$: \[ -\frac{f(x_{n-1})}{f'(x_{n-1})} = x_n - x_{n-1} \] Rearranging terms to solve explicitly for $x_n$: \[ x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})} \] This matches Option (A). Let us briefly clarify alternative option descriptions: - Option (B) represents a structure similar to Euler's method for differential expressions. - Option (D) corresponds to the Secant method formula written recursively.